We have seen that we can control the closcc-loop gai,n v..Ivu, of an operational [amplifier by introducing feedback through external resistor: combinations. We wi h now to examine the feedback mechanism in detail and discover some other important consequences of its use. Feedback theory is widely used to study the behavior of electronic components as wel! a complex systems in many if erent technical fields so it is important to develop an appreciation and undersrinding of it’ underlying prlnciples.

**FeedbacK in the Noninverting Amplifier**

We begin our study of feedback principles with an analysis of the’ noninvcrting amplifier. Figure \3-8 shows that configuration along ‘with un equivalent block ~iagram on which we can iden i:f the signal and feedback paths. The block labeled II represents the amplifier and its open-loop gain, and the block labeled f3 is the reedbuck path. The quantity {3is en!’ l the feet/hack ratio and represents the portion of the output voltage that is fed kh..k to t e input. For example, if f3 = n.s, then a voltage equal to one-half the output level is fed back to the input”. Notice the special symbol where the input and feedbacx paths come together. This syinbol represents the differential action at the amplifier’s input. It is usually edited a summing j I ction, although in our case it is performing (\ differencing operation. as indicated by the + and – symbols. The output nf the junction, which is the input to the amplifier. is seen to he V” = Vi” – vf. V.’ is often called the error voltage. Note that it corresponds to Vi” – vi in the noninverting amplifier, and under ideal conditions is equal to 0 (equation 13-13). The feedback voltage vf = /3v .• corresponds to u; in the amplifier circuit. Since the feedback voltage subtracts from the input voltage, the amplifier is said to have negative feedback.

Equation 13-18 is a very important and very useful result. If expresses the closedloop gain v..IvII/ as a function of the open-loop gain A and the feedback ratio (3. We can now-apply this result to the noninvcrting amplifier, in Figure 13-8(a), Notice that R, and RI form a voltage divider across the output of the amplifier,

Equation 13-18 shows that the closed-loop gain of a real amplifier also departs from that of an ideal amplifier when the value of f3 becomes very small. Small values PI’ {3correspond to large closed-loop gains.

The last two examples have shown that the closed-loop gain departs from the ideal value of 1I{3 when A is small or when {3 is small. We can deduce that fact from another examination of equation 13-18:

Clearly both A and {3should be large if we want vJv,” to equal 1I{3. The product Af3 is called the loop gain and is very useful in predicting the behavior of a feedback system. The name loop gain is derived from its definition as the product of the gains in the feedback model as one travels around the loop from amplifier input. through the amplifier. and through the feedback path (with the summing junction open). •

Negative feedback improves the performance of an amplifier in several ways. In the ease of the noninverting ampliflcr, it can be shown that the input resistance seen by the signal snurcc (looking directly into the + terminal)

where 1′,,/ is the differential input resistance of the amplifier. This equation shows that the input resistance is rid multiplied by the factor 1 + A~, which is usually much greater than I and is approximately equal to the loop gain A{3. For example. if rid == ~o kn. A == 10′. and {3= 0.01, then /’11I ~ (20 X 10′)(10′)(0.01) = 20 MO, a very respectable value. In the case of the voltage follower, {3== 1, and ru is multiplied by the full value of A, which accounts for the extremely large input resistance it can provide in buffer applications.

The closed-loop output resistance of the noninverting amplifier is also improved by negative feedback

where 1′” is the open-loop output resistance of the amplifier. Equation 13-24 shows that the output resistance is decreased by the same factor by which the.Innut resistance is increased. A typical value for r; is 75 0, so with A = IOsand {3″” I}I

we have r,,(stage) “= 75/m’ = 0.075 n. which is very close to the idc.d value ofO. Finally. negative feedback reduces distortion causeJ by the amplilier itself. We will study this property later, in connection with large-signal amplifiers.